A priori estimates for solutions to a class of obstacle problems under p, q-growth conditions

Gavioli, Chiara

doi:10.1007/s41808-019-00043-y

Cited by 22 publications

(13 citation statements)

References 28 publications

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“…holds for all balls B R/8 ⊂ B R/2 ⊂ B R ⋐ Ω, with C = C(ν, L, q, p, r, n, R) and α = α(p, q, r, n), where we defined Once estimate (3.2) is estabilished, by applying a suitable approximation procedure (see [29] for the details) we get the following corollary.…”

Section: Preliminary Resultsmentioning

confidence: 87%

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Regularity for Obstacle Problems without Structure Conditions

Bertazzoni¹,

Riccò²

2021

Preprint

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This paper deals with the Lipschitz regularity of minimizers for a class of variational obstacle problems with possible occurance of the Lavrentiev phenomenon. In order to overcome this problem, the availment of the notions of relaxed functional and Lavrentiev gap are needed. The main tool used here is a ingenious Lemma which reveals to be crucial because it allows us to move from the variational obstacle problem to the relaxed-functional-related one. This is fundamental in order to find the solutions' regularity that we intended to study. We assume the same Sobolev regularity both for the gradient of the obstacle and for the coefficients.G. BERTAZZONI, S. RICC Ò So, the aim of this work is to complement the results cointained in the paper [11], where authors obtain Lipschitz regularity results for obstacle problems with Sobolev regularity for the coefficients and where the lagrangian f satisfies p, q−growth conditions without assuming that Lavrentiev phenomenon may occur. This phenomenon is a clear obstruction to regularity, since (1.2) prevents minimizers to belong to W 1,q . Notice that (1.2) cannot happen if p = q or if f is autonomous (it not depends on variable x) and convex. Moreover, as pointed out in Section 3 of [22], the appearance of (1.2) has geometrical reasons and cannot be spotted in a direct way by standard elliptic regularity techniques. Therefore, the basic strategy in getting regularity results consists in excluding the occurrence of (1.2) by imposing that the Lavrentiev gap functional L(u), defined in (2.10), vanishes on solutions. However, here in this manuscript we adopt a different viewpoint, following the lines of [20].We present a general Lipschitz regularity result by covering the case in which the Lavrentiev phenomenon may occur. In this respect, a key role will be played by the relaxed functional.We therefore need to introduce the exact framework of relaxation in the case of obstacle problems and then we will state our main result. The crucial step will be constituted by Lemma 5.1 which is the natural counterpart of the necessary and sufficient condition to get the absence of Lavrentiev phenomenon.We will state and prove the result with Sobolev dependence on both the obstacle and the partial map x → D ξ f (x, ξ). A model functional that is covered by our results isdx with q > p > 1 and a(•) a bounded Sobolev coefficient. The plan of the paper is the following: in Section 2 we state our model problem and the main results of the paper, in Section 3 we present some preliminary results we need in the sequel; Section 4 is devoted to the presentation of our a-priori estimate and finally in Section 5 we present the Lipschitz regularity results for solutions to the relaxed obstacle problem.

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Section: Preliminary Resultsmentioning

confidence: 87%

“…Our hypothesis implies the ones needed for this theorem to be true. The proof can be found in [29]. Theorem 3.2.…”

Section: Preliminary Resultsmentioning

confidence: 98%

Regularity for Obstacle Problems without Structure Conditions

Bertazzoni¹,

Riccò²

2021

Preprint

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“…, this is the crucial point where the Calderón-Zygmund result is used. We observe that for instance in the paper [17] this problem has been overcome by means of the results in [13] which provides the necessary higher integrability result so this term turns to be bounded. Since Dψ ∈ W 1,p + (Ω), classical Sobolev embedding Theorem implies Dψ ∈ L (p + ) * (Ω).…”

Section: Proof Of Theorem 22mentioning

confidence: 97%

“…Here Ω is a bounded open set of R n , n > 2 and F : Ω × R n → R is a Carathéodory function fulfilling natural growth and convexity assumptions with variable exponent (namely assumptions (A1)-(A3) below). Higher diferentiability results have been attracting a lot of attention in the recent years, starting from the pioneering papers [32,33,34,21,22,29], to the more recent results concerning higher differentiability results for obstacle problems in the case of standard growth conditions [13,14,24] and p − q growth conditions of integer [17,16,6] and fractional order [25], see also [6], including the case of nearly linear growth [18] and the subquadratic growth case [19]. In the same spirit of these results, assuming that the gradient of the obstacle belongs to a suitable Sobolev class, we are interested in finding conditions on the partial map x −→ A(x, ξ) := D ξ F (x, ξ) in order to obtain that the extra differentiability property of the obstacle transfers to the gradient of the solution, possibly with no loss in the order of differentiability.…”

Section: Introductionmentioning

confidence: 99%

Higher differentiability of solutions for a class of obstacle problems with variable exponents

Foralli¹,

Giliberti²

2021

Preprint

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“…Here, as we already said, we are interested in higher differentiability results since in case of non standard growth, many questions are still open. In [6,7,12,17,18,21,27,31,38] the authors analyzed how an extra differentiability of integer or fractional order of the gradient of the obstacle provides an extra differentiability to the gradient of the solutions, also in case of standard growth. However, since no extra differentiability properties for the solutions can be expected even if the obstacle ψ is smooth, unless some assumption is given on the x-dependence of the operator A, the higher differentiability results for the solutions of systems or for the minimizers of functionals in the case of unconstrained problems (see [1,8,10,19,20,22,23,24,36,37]) have been useful and source of inspiration also for the constrained case.…”

Section: Introductionmentioning

confidence: 99%

Regularity results for bounded solutions to obstacle problems with non-standard growth conditions

Gentile¹,

Giova²,

Torricelli³

2021

Preprint

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In this paper we consider a class of obstacle problems of the typewhere ψ is the obstacle, K ψ (Ω) = {v ∈ u0 + W 1,p 0 (Ω, R) : v ≥ ψ a.e. in Ω}, with u0 ∈ W 1,p (Ω) a fixed boundary datum, the class of the admissible functions and the integrand f (x, Dv) satisfies non standard (p, q)-growth conditions. We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map x → A(x, ξ) is independent of the dimension n and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one W 1,n .

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A priori estimates for solutions to a class of obstacle problems under p, q-growth conditions

Cited by 22 publications

References 28 publications

Regularity for Obstacle Problems without Structure Conditions

Regularity for Obstacle Problems without Structure Conditions

Higher differentiability of solutions for a class of obstacle problems with variable exponents

Regularity results for bounded solutions to obstacle problems with non-standard growth conditions

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